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Numerade Educator

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Problem 17 Easy Difficulty

Find the general indefinite integral.

$ \displaystyle \int 2^t (1 + 5^t)\, dt $

Answer

$\frac{2^{t}}{\ln 2}+\frac{10^{t}}{\ln 10}+C$

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Video Transcript

right. The theme in this problem is that if you are asked to take a to the T. Power D. T. So the location of your expense important, uh The anti griddle is one divided by the natural log of T. Um times A. To the T. Power. And it's just undoing the chamber and you do need to remember plus your constant. So the key in this problem then is to get your problem to look like that uh to the T. One plus five to the T. Power DT. So what we need to do is because there's a multiplication right here, we need to distribute that in there before trying anything. And a rule here is that because we have to the T. Power. So there's T. Number of twos and there's T. Number of fives. That should be the same as two times 52 times five tee times. So it's the same thing as 10 to the T. Power. I hope my explanation made sense why that's true because now we can just use this rule up here. Uh put a boxer on that, that's what we're using. Um And it's going to be won over natural log of two times, two to the T. Power, plus one over natural log of 10 times 10 to the T. Power. And don't forget about plus C. You can verify that this is correct because the derivative of this will get us back to here, which we already know is equal to that, so we have the correct answer.